A037276 -id:A037276 - cp's OEIS frontend

A048985 Working in base 2, replace n with the concatenation of its prime divisors in increasing order (write answer in base 10).

1, 2, 3, 10, 5, 11, 7, 42, 15, 21, 11, 43, 13, 23, 29, 170, 17, 47, 19, 85, 31, 43, 23, 171, 45, 45, 63, 87, 29, 93, 31, 682, 59, 81, 47, 175, 37, 83, 61, 341, 41, 95, 43, 171, 125, 87, 47, 683, 63, 173, 113, 173, 53, 191, 91, 343, 115, 93, 59, 349, 61, 95, 127, 2730

Offset: 1

N. J. A. Sloane

			15 = 3*5 -> 11.101 -> 11101 = 29, so a(15) = 29.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Patrick De Geest, Home Primes

Cf. A037276, A048986, A064841.
Cf. A193652, A029744 (record values and where they occur).
Cf. A027746.

-- import Data.List (unfoldr)
a048985 = foldr (\d v -> 2 * v + d) 0 . concatMap
   (unfoldr (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2))
   . reverse . a027746_row
-- Reinhard Zumkeller, Jul 16 2012

f[n_] := FromDigits[ Flatten[ IntegerDigits[ Flatten[ Table[ #1, {#2}] & @@@ FactorInteger@n], 2]], 2]; Array[f, 64] (* Robert G. Wilson v, Jun 02 2010 *)

from sympy import factorint
def a(n):
    if n == 1: return 1
    return int("".join(bin(p)[2:]*e for p, e in factorint(n).items()), 2)
print([a(n) for n in range(1, 65)]) # Michael S. Branicky, Oct 07 2022

More terms from Sam Alexander (pink2001x(AT)hotmail.com) and Michel ten Voorde

A248915 Composite numbers which divide the concatenation of their prime factors, with multiplicity, in descending order.

Original entry on oeis.org

378, 12467, 95823, 10715274, 13485829, 111495095, 42002916561, 176685987695

Offset: 1

Paolo P. Lava, Oct 16 2014

Prime numbers are not considered because they trivially satisfy the relation.
For terms in ascending order see A259047 and StackExchange link. [Paolo P. Lava, May 30 2019]
a(9) <= 3953318131772867. - Chai Wah Wu, Apr 12 2024
a(2), the bound for a(9) above, and larger terms may be found using an extension of Andersen's algorithm to arbitrary base and ordering (see links for an implementation and another term). - Michael S. Branicky, Apr 13 2024

			Prime factors of 378 are 2,3,3,3,7; concat(7,3,3,3,2) = 73332 and 73332/378 = 194.

Michael S. Branicky, Python program for Andersen's algorithm extended to arbitrary base/ordering
StackExchange, New term in ascending order

Cf. A037276, A069872, A224930, A240265, A259047.

with(numtheory); P:=proc(q) local a,b,c,d,j,k,n;
for n from 1 to q do if not isprime(n) then a:=ifactors(n)[2]; b:=[]; d:=0;
for k from 1 to nops(a) do b:=[op(b),a[k][1]]; od; b:=sort(b);
for k from nops(a) by -1 to 1 do c:=1; while not b[k]=a[c][1] do c:=c+1; od;
for j from 1 to a[c][2] do d:=10^(ilog10(b[k])+1)*d+b[k]; od; od;
if type(d/n,integer) then print(n); fi;
fi; od; end: P(10^9);

isok(n) = {my(s = ""); my(f = factor(n)); forstep (i=#f~, 1, -1, for (k=1, f[i,2], s = concat(s, Str(f[i,1])))); (eval(s) % n) == 0;} \\ Michel Marcus, Jun 16 2015

a(7)-a(8) from Giovanni Resta, Jun 16 2015

A006919 Write down all the prime divisors in previous term.

Original entry on oeis.org

8, 222, 2337, 31941, 33371313, 311123771, 7149317941, 22931219729, 112084656339, 3347911118189, 11613496501723, 97130517917327, 531832651281459, 3331113965338635107, 3331113965338635107

Offset: 1

N. J. A. Sloane

H. Jaleebi, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Patrick De Geest, Home Primes

Cf. A056938 (same for a(1)=49), A037271-A037276, A048985, A048986, A049065.

g[ n_ ] := (x = n; d = {}; While[ FactorInteger[ x ] != {}, f = FactorInteger[ x, FactorComplete -> True ][ [ 1, 1 ] ]; x = x/f; AppendTo[ d, IntegerDigits[ f ] ] ]; FromDigits[ Flatten[ d ] ]); NestList[ g, 8, 15 ]
NestList[FromDigits[Flatten[IntegerDigits/@(Table[First[#],{Last[#]}]& /@ FactorInteger[#])]]&,8,15] (* Harvey P. Dale, Dec 04 2011 *)

first(N, a=8)=vector(N,i,if(i>1,a=A037276(a),a)) \\ M. F. Hasler, Oct 07 2022

a(n+1) = A037276(a(n)), a(1) = 8. - M. F. Hasler, Oct 07 2022

More terms from Robert G. Wilson v, Sep 05 2000, who remarks that sequence stabilizes at 13th term with a prime.

A073646 Least number formed by concatenating the prime factors of n in base 10.

Original entry on oeis.org

1, 2, 3, 22, 5, 23, 7, 222, 33, 25, 11, 223, 13, 27, 35, 2222, 17, 233, 19, 225, 37, 112, 23, 2223, 55, 132, 333, 227, 29, 235, 31, 22222, 113, 172, 57, 2233, 37, 192, 133, 2225, 41, 237, 43, 1122, 335, 223, 47, 22223, 77, 255, 173, 1322, 53, 2333, 115, 2227

Offset: 1

Reinhard Zumkeller, Aug 29 2002

a(n)<=A037276(n) for all n and a(n)=A037276(n) for n<22, a(22) = a(2*11) = 112 <> A037276(22) = 211.

			a(6) = a(2*3) = 23 < 32; a(66) = a(2*3*11) = 1123 < 1132 < 2113 < 2311 < 3112 < 3211.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Different from A037276. Cf. A084797.

import Data.List (sort)
a073646 :: Integer -> Integer
a073646 = read . concat . sort . map show . a027746_row
-- Reinhard Zumkeller, Jul 13 2013

con[n_, k_] := Nest[Join[{#}, {n}] &, n, k - 1]; Table[Min[FromDigits /@ Flatten /@ IntegerDigits[Permutations[Flatten[con @@@ FactorInteger[n]]]]], {n, 56}] (* Jayanta Basu, Jul 04 2013 *)

Sort {A027746(n,k): k=1..A001222(n)} lexicographically and concatenate. - Reinhard Zumkeller, Jul 13 2013

A106554 Concatenation of the two prime divisors of a semiprime, smallest divisor first.

Original entry on oeis.org

22, 23, 33, 25, 27, 35, 37, 211, 55, 213, 311, 217, 57, 219, 313, 223, 77, 317, 511, 319, 229, 231, 513, 323, 237, 711, 241, 517, 243, 329, 713, 331, 247, 519, 253, 337, 523, 259, 717, 1111, 261, 341, 343, 719, 267, 347, 271, 1113

Offset: 1

Eric Angelini, May 09 2005

Concatenation of the divisors starting with the largest one leads to another sequence.

			First semiprime is 4; 4 is 2*2 -> 22.
Second semiprime is 6; 6 is 2*3 -> 23.
Third semiprime is 9; 9 is 3*3 -> 33.
Fourth semiprime is 10; 10 is 2*5 -> 25.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A106550, A084126, A084127, A001358.

read("transforms") :
A106554 := proc(n)
    A037276(A001358(n)) ;
end proc: # R. J. Mathar, Oct 29 2012

FromDigits@ Flatten[IntegerDigits@ Table[#1, {#2}] & @@@ FactorInteger@ #] & /@ Select[Range@ 144, PrimeOmega@ # == 2 &] (* Michael De Vlieger, Oct 01 2015 *)

do(x)=my(v=List()); forprime(p=2, sqrt(x), forprime(q=p, x\p, listput(v, [p*q, eval(Str(p, q))]))); Vec(apply(u->u[2], vecsort(v, 1))) \\ Charles R Greathouse IV, Sep 30 2015

a(n) = A037276(A001358(n)). - R. J. Mathar, Oct 29 2012

More terms from Reinhard Zumkeller, May 19 2005

Data corrected by Giovanni Teofilatto and Altug Alkan, Oct 01 2015

A340592 a(n) is the concatenation of the prime factors (with multiplicity) of n mod n.

Original entry on oeis.org

0, 0, 2, 0, 5, 0, 6, 6, 5, 0, 7, 0, 13, 5, 14, 0, 17, 0, 5, 16, 13, 0, 15, 5, 5, 9, 3, 0, 25, 0, 14, 14, 13, 22, 1, 0, 29, 1, 25, 0, 27, 0, 11, 20, 39, 0, 47, 28, 5, 11, 29, 0, 11, 16, 43, 34, 55, 0, 15, 0, 45, 22, 14, 58, 1, 0, 41, 47, 47, 0, 57, 0, 15, 55, 15, 18, 51, 0, 65, 12, 77, 0, 53, 7

Offset: 2

J. M. Bergot and Robert Israel, Jan 12 2021

a(n) = 0 if n is prime.
The first composite n for which a(n)=0 is 28749. Are there others?
There are no other composite n terms for which a(n)=0 up to 5 million. - Harvey P. Dale, Jul 17 2023

			For n = 20 = 2*2*5, a(20) = 225 mod 20 = 5.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A037276, A340594, A340595.

dcat:= proc(L) local i,x;
  x:= L[-1];
  for i from nops(L)-1 to 1 by -1 do
    x:= 10^(1+ilog10(x))*L[i]+x
  od;
  x
end proc:
f:= proc(n) local F;
  F:= sort(ifactors(n)[2],(a,b) -> a[1] < b[1]);
  dcat(map(t -> t[1]$t[2], F)) mod n;
end proc:
map(f, [$2..100]);

Table[Mod[FromDigits[Flatten[IntegerDigits/@Table[#[[1]],#[[2]]]&/@FactorInteger[n]]],n],{n,2,100}] (* Harvey P. Dale, Jul 17 2023 *)

from sympy import factorint
def a(n):
    if n == 1: return 0
    return int("".join(str(f) for f in factorint(n, multiple=True)))%n
print([a(n) for n in range(2, 86)]) # Michael S. Branicky, Jan 18 2022

a(n) = A037276(n) mod n.

A251360 Numbers k such that k is the concatenation of prime factors of pi(k), in increasing order.

Original entry on oeis.org

1117, 2163, 2537, 5137, 222926801

Offset: 1

Jahangeer Kholdi, Dec 01 2014

Numbers k such that k = A037276(A000720(k)).
Conjecture: numbers k such that k = A084317(A000720(k)). - Chai Wah Wu, Apr 04 2018
a(6) > 10^12 if exists. - Max Alekseyev, May 16 2025

			1117 is in the sequence since pi(1117) = 11*17,
2163 is in the sequence since pi(2163) = 2*163,
2537 is in the sequence since pi(2537) = 2*5*37,
and 5137 is in the sequence since pi(5137) = 5*137.

Chris Caldwell, G. L. Honaker and Lewis, 1117

Cf. A000040, A000720, A251361, A251362.

a251360[n_Integer] := Select[Range[n], # == FromDigits[Flatten@IntegerDigits[First@ Transpose@ FactorInteger[PrimePi[#]]]] &]; a251360[10^5] (* Michael De Vlieger, Dec 03 2014 *)

from sympy import prime, factorint
A251360_list, p = [], 3
for n in range(2,10**6):
    q, fn = prime(n+1), factorint(n)
    m = int(''.join(str(d)*fn[d] for d in sorted(fn)))
    if p <= m < q:
        A251360_list.append(m)
    p = q # Chai Wah Wu, Dec 10 2014, corrected Apr 04 2018

a(5) from Chai Wah Wu, Dec 10 2014

A343156 Starting at n, a(n) = number of iterations of the map x -> A084317(x) (concatenate distinct prime factors of x) required to reach a prime, or -1 if no prime is ever reached.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 2, 4, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 4, 1, 2, 0, 2, 0, 1, 1, 5, 3, 1, 0, 2, 1, 2, 0, 2, 0, 1, 4, 1, 0, 1, 1, 2, 1, 4, 0, 1, 2, 2, 2, 1, 0, 2, 0, 3, 1, 1, 3, 1, 0, 5, 3, 1, 0, 1, 0, 2, 4, 2, 2, 2, 0, 2, 1, 1, 0, 2, 3, 2, 3, 1, 0, 2, 64, 1, 1, 2, 4, 1, 0, 2, 1, 2

Offset: 2

N. J. A. Sloane, Apr 07 2021

Judging by the behavior of similar sequences, it is likely that almost all values of a(n) are -1. n = 407 (see A343157) seems to be the first open case.

			10 = 2*5 -> 25 = 5^2 -> 5, prime, taking two steps, so a(10)=2.
a(91) = 64: see A084319.

Eric Angelini, W. Edwin Clark, Hans Havermann, Frank Stevenson, Allan C. Wechsler, and others, Postings to Math Fun mailing list, April 2021.

Hans Havermann, Table of n, a(n) for n = 2..406

Cf. A084317, A037274, A037276, A084319, A195264, A343157.

See A343158 for when k first appears.

A371696 Composite numbers that divide the reverse of the concatenation of their ascending order prime factors, with repetition.

Original entry on oeis.org

26, 38, 46, 378, 26579, 84941, 178838, 30791466, 39373022, 56405502, 227501395, 904085931, 1657827142

Offset: 1

Scott R. Shannon, Apr 03 2024

a(14) > 10^10, if it exists. - Daniel Suteu, Apr 28 2024

			26 is a term as 26 = 2 * 13 = "213" which is reverse is "312", and 312 is divisible by 26.
227501395 is a term as 227501395 = 5 * 11 * 17 * 23 * 71 * 149 = "511172371149" which in reverse is "941173271115", and 941173271115 is divisible by 227501395.

Cf. A027746, A037276, A259047, A371641, A371666.

from itertools import count, islice
from sympy import isprime, factorint
def ok(k): return not isprime(k) and int("".join(str(p)[::-1]*e for p, e in list(factorint(k).items())[::-1]))%k == 0
def agen(): yield from filter(ok, count(4))
print(list(islice(agen(), 7))) # Michael S. Branicky, Apr 13 2024

from itertools import count, islice
from sympy import factorint
def A371696_gen(startvalue=4): # generator of terms >= startvalue
    for n in count(max(startvalue,4)):
        f = factorint(n)
        if sum(f.values()) > 1:
            c = 0
            for p in sorted(f,reverse=True):
                a = pow(10,len(s:=str(p)),n)
                q = int(s[::-1])
                for _ in range(f[p]):
                    c = (c*a+q)%n
            if not c:
                yield n
A371696_list = list(islice(A371696_gen(),6)) # Chai Wah Wu, Apr 13 2024

a(13) from Daniel Suteu, Apr 28 2024

A038526 Concatenation of prime factors of n-th Fibonacci number.

Original entry on oeis.org

2, 3, 5, 222, 13, 37, 217, 511, 89, 222233, 233, 1329, 2561, 3747, 1597, 2221719, 37113, 351141, 213421, 89199, 28657, 2222233723, 553001, 233521, 21753109, 31329281, 514229, 2225113161, 5572417, 37472207, 28919801, 15973571, 513141961, 22223331719107, 731492221

Offset: 3

Jeff Burch

Alois P. Heinz, Table of n, a(n) for n = 3..380

A055642 := proc(n) if n =0 then 1 ; else ilog10(n)+1 ; fi ; end: catDigs := proc(L) local a,k,i ; a := 0 ; for i from 1 to nops(L) do a := a*10^A055642(L[i])+L[i] ; od ; end: A037276 := proc(n) local L,i,ifs,j ; if n = 1 then 0 else ifs := ifactors(n)[2] ; L := [] ; for i in ifs do for j from 1 to op(2,i) do L := [op(L),op(1,i)] ; od: od: catDigs(L) ; fi ; end: A038526 := proc(n) A037276(combinat[fibonacci](n)) ; end: seq(A038526(n),n=3..20) ; # R. J. Mathar, Oct 12 2007
# second Maple program:
F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
a:= n-> parse(cat(sort(map(i-> i[1]$i[2], ifactors(F(n))[2]))[])):
seq(a(n), n=3..40);  # Alois P. Heinz, Aug 17 2018

a(n) = A037276(A000045(n)). - R. J. Mathar, Oct 12 2007

Edited by Charles R Greathouse IV, Apr 27 2010

cp's OEIS Frontend

A048985 Working in base 2, replace n with the concatenation of its prime divisors in increasing order (write answer in base 10).

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A248915 Composite numbers which divide the concatenation of their prime factors, with multiplicity, in descending order.

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A006919 Write down all the prime divisors in previous term.

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A073646 Least number formed by concatenating the prime factors of n in base 10.

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A106554 Concatenation of the two prime divisors of a semiprime, smallest divisor first.

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A340592 a(n) is the concatenation of the prime factors (with multiplicity) of n mod n.

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A251360 Numbers k such that k is the concatenation of prime factors of pi(k), in increasing order.

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